The $p$-adic Number System and the Artin-Hasse Exponential

Achyut Bharadwaj, Lex Harie Pisco, Krittika Garg, Swayam Chaulagain, Counsellor: Sanskar Agrawal, Mentor: Nischay Reddy
June 2023
$p$-adic numbers, Artin-Hasse exponential, radius of convergence, power series, integral power series, discs, leftist numbers

Introduction # In this paper, we explore the $p$-adic system, by defining it in multiple ways: as an extension of the $p$-adic integers, as well as an extension of the rationals. We then proceed to perform analysis in the $p$-adics, by defining convergence, continuity and discs. We then describe exponentiation and logarithmic functions over the $p$-adics as functions derived from power series. We explore the radius of convergence and other properties of these functions. ...